Inner and outer isometries of graphs of polynomials

@misc{fernandes2026inner,
  title = {{Inner and outer isometries of graphs of polynomials}},
  author = {Alexandre Fernandes and Claudia Fernandes and Edson Sampaio},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14291},
  eprint = {2610.14291},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14291},
}

TY  - UNPB
TI  - Inner and outer isometries of graphs of polynomials
AU  - Alexandre Fernandes
AU  - Claudia Fernandes
AU  - Edson Sampaio
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.14291
UR  - https://arxiv.org/abs/2610.14291
AB  - We classify graphs of complex polynomials in one variable up to isometry, both for the inner (length) distance and for the outer (Euclidean) distance of ℂ^2≅ℝ^4. For polynomials p,q with °p≥ 2 we show that the graphs of p and q are inner isometric if and only if they are outer isometric, if and only if p(z)=α q(βz+h)+k or p(z)=α q̅(βz+h)+k for some α,β,h,k∈ℂ with |α|=|β|=1, where q̅ denotes the polynomial with conjugate coefficients. Moreover, every inner isometry between such graphs is the restriction of an isometry of ℂ^2; in this sense graphs of complex polynomials are isometrically rigid. We contrast this with the real case, where the two distances behave very differently: the graphs of any two real polynomials are inner isometric, whereas, for degree at least $2$, outer isometries are generated by translations, reflections in horizontal and vertical lines and half-turns.
ER  -